Saturday, February 9, 2013

There
are two known sins that one can commit in relation to books: a) to
prematurely judge a book by its cover; and b) to believe in the
tempting correlation that a short book can be read in a short amount
of time.

I
have managed to stay clear from the first bookish sin since the name
of the author has more value to me than its graphic representations,
but in the case of David Berlinski's The King of Infinite Space:
Euclid and his Elements I made the error of thinking that its
slimness, the relatively and comparatively few number of pages meant
that I could finish it in a jiffy. Not so. I learned my (mathematical)
lesson the cumbersome way because a lot of complex information can be
packed in small parcels of space.

Entering
the world of Euclid, as imagined by Berlinski, is like sitting in a
wide-angled out-of-focus high school classroom where the bespectacled
math teacher is hardly audible and the clock keeps receding instead
of advancing in time. Or so was my own general feeling when I
attempted to read and re-read this book.

I
must make a clear confession before I go on: I lost my math ability
and most enjoyment thereof after age 17. It happened overnight and
there is, in fact, no reasonable explanation for it. Math simply
vanished out of my mind, and I carefully circumvented and kept out of
its shadows and surroundings whenever I could.

Fortunately,
math and literature are not the closest of friends, so I have been
safe for most of my life. But in philosophy you will stumble upon
math every once in a while, while the fascinating and overwhelming
world of quantum mechanics will take even seasoned mathematicians for
a hearty spin.

I
cannot deny the fact that I envy mathematicians out there: they have
access to a world that I will never understand. The Greek double
Pythagoras and Plato were great mathematician-philosophers, so were
their French double counterparts Descartes and Pascal. And each and
everyone had powerful things to say about human nature using math as
a their guiding light.

Reading
about Euclid's theorems and axioms and postulates re-vibrated my
Platonic sensibilities. I realized that both Plato and Euclid either
lived in an elaborate world of fantasy or that they correctly
imagined the existence of a world outside of this world, one that
defied imagination and general conventions. They indeed spoke and
were inspired by the same abstract and grammatically precise language
of mathematics.

As
I was walking home one day immersed in Euclidean thoughts I could not
help feeling slightly overwhelmed by the world of geometry myself. In
fact, everything is geometrical! Buildings construct a right angle to
the ground and stretch out in an imaginary infinite line through the
sky; streets are straight lines that are parallel to each other; shop
signs and ladders on the street form triangles. You cannot escape
shapes; even my face and pot belly are round-shaped and my hanging arms are
straight lines.

And
yet, it turns out that Euclid needed to be modified when it comes to spheres and the physical
world because the shortest distance between two points is not always
a straight line. Curves and geodesics within spheres, gravity and elliptical forces make it not so
what on paper looks perfectly sound and fine. And the sum of the
angles of triangles is not always 180 degrees in real life.

And
then it came to me. This is what Plato must have meant by the
heavenly forms! Yes, there must be a perfect triangle out there, one
that fully represents what is drawn on two-dimensional space.
Whatever shape we encounter may then be an imperfect replication of
its heavenly and otherworldly ideal. We live in a non-Euclidean world
because it is the shadow of the infinite space that was imagined by
Plato and reformulated by Euclid.

Oh,
my God! And it had seemed that this book by mathematician David
Berlinski was a bore or drudgery! Not so if math is your cup of tea
of course, but to me all the abstract logical talk had given me a
slight headache. Until the information somehow seeped in into my
subconscious, I suppose.

Euclid
spent a lot of time to prove his theorems with his various
definitions and postulations. Why did he take such pains, I asked
myself? And yet, it was this strategy of taking pains to try to
convince others through logic that has shaped the Western
consciousness. Socrates and Plato had used reason to debate points
and to persuade and enlighten others. Euclid used geometry. Descartes
applied a similar mathematical approach to the methodology of
philosophy itself.

And
Euclid's ideas had withstood time over more than two thousand years
(compare that with the nimbly two hundred years or so that Newton's
mechanical outlook lasted!) until 19th century
mathematicians turned it upside down, and Einstein put the proverbial
dagger through the hearts of Euclid and Newton respectively.

And
yes, the very foundations of the Earth had begun to shake! Berlinski
quotes how Bertrand Russell and others were shocked at the opening
cracks in a once stable foundation of mathematical precision.
Suddenly nothing was certain anymore and everything became relative.
And the problem with relativity is that there is no absolute truth or
theorems, no clear formulas.

Plato
will say I told you so! He will insist on his version of another
idealized Euclidean plane of existence where triangles are as they
should be, namely perfect. Life can become more predictable again; we
can calculate shapes and movements without elements interfering with
our studies. Put differently, it is not the triangle that is wrong;
it is our faulty perception of it.

Of
course, had I more knowledge about Euclid or the subject matter I
would have gotten more out of this book. As an afterword under his
Teacher's Note, Berlinski talks about Euclid's book, something that could be easily applied to his own work:
“The book demands both effort and concentration. The proofs do not
come easy.”

Although
there is humor to help us along, it comes as dry as unsalted
crackers and the language is quite sparse. Where I would expect to be
taking off on flights of fancy, there is but a cracked line. It is
all as geometrically efficient as it gets.

At
first sight, the book seems to have absolutely nothing to do with
one's own life being as abstract as it can be, but it is not unlike
Euclid's work itself: Dig in there for a while, and you will find
something worth your time and effort.

3 comments:

You make some sweeping high-level assertions here. How can you say Euclid was wrong? The existence of non-Euclidean geometry alongside doesn't make Euclid wrong, just as Newtonian mechanics is not made wrong by Einstein's theorems etc which apply to other scales of magnitude, limiting cases and so on.

And I well remember when I was first taught geometry, probably from a translation of Euclid, for we had old-fashioned books. We were taught straight away that a point has no dimension at all, so cannot physically exist, and a line has length but no width, so cannot be physically drawn, etc.

One thing was very easy to understand: that the shortest difference between two points is (provably) a straight line. We were also told about the curvature of the earth, the inadequacies of the Mercator's Projection (used for our wall map of the world) and that the most direct aeroplane journeys are segments of Great Circles (circles whose centre is the centre of the earth).

With respect, Plato never came into it! But when we did find out about him, his ideas seemed rather Christian. (Doubtless because his philosophy did influence Christianity.)

Well, I have given the guy credit since he has been right for two thousand years! That's a really long time.

And I do not imply that he is "wrong" in an Earth-is-flat-kind-of-way, a binary either with us or against us outlook.

Newton was the absolute norm until we realized that it was not a perfect encompassing model. So we had to readjust his theory. The same happened to Euclid. Neither is wrong necessarily; they are not completely right either.

And this could lead to a more philosophical discussion of what we mean as right or wrong in the first place and where what applies.

Unfortunately my math background will not be able to suffer such stretches. And most likely I have myself solely to blame for this because I am sure that my high school math teachers were rather competent.

Dear Vincent, I stand corrected and my sincere apologies to you, to all high school math teachers and particularly to Sir Euclid himself.

After an enlightening and instructive chat with a respected mathematician I decided to change a few lines here and there in order to soften the margin of error. It turns out that Euclid was not necessarily wrong as I had myself erroneously and somewhat prematurely claimed.

But I still stand by the claim that Newton was not fully right as his theory was not as absolute as it was thought to be in a pre-Einsteinian world.